Integral operators with two-sided cusp singularities (Q2708380)

The authors continue the study of the relation of properties of Fourier integral operators \({\mathcal F}:{\mathcal C}_{\text{comp}}(Y)\to {\mathcal C}'(X)\) to the geometry of the projections \(\pi_{\text{L}}: C\to T^*X\) and \(\pi_{\text{R}}:C\to T^*Y\) from the canonical relation \(C\subset (T^* X(0)\times (T^*Y\setminus 0)\) associated to \({\mathcal F}\). They consider the case when both \(\pi_{\text{L}}\) and \(\pi_{\text{R}}\) have cusp singularities, that is, when each of them can be represented in certain local coordinates by the form NEWLINE\[NEWLINE(x_1,\dots, x_m)\mapsto (y_1= x_1,\dots, y_{m-1}= x_{m-1}, y_m= n^3_m- x_1x_m).NEWLINE\]NEWLINE This case includes Radon transforms associated to generic families of curves in \(\mathbb{R}^4\).NEWLINENEWLINENEWLINEThe main result is an \(L^p\)-continuity theorem of \({\mathcal F}\). In particular, for \(p=2\), let \(\dim X=\dim Y= n\), and let \({\mathcal F}\in I^\mu(X,Y,C)\) be a Fourier integral operator of order \(\mu\), associated to the canonical relation \(C\). Assume that both projections \(\pi_L\) and \(\pi_R\) have at most cusp singularities. Then \({\mathcal F}\) extends to a continuous operator from \(L^2_{\alpha,\text{comp}}(Y)\) to \(L^2_{\beta,\text{loc}}(X)\) if \(\mu< \alpha-\beta-1/4\).NEWLINENEWLINENEWLINEThe proof is based on the analysis of an oscillatory integral operator depending on a parameter \(\lambda\gg 1\).